Hybrid subdivision in computer graphics

ABSTRACT

Different limit surfaces are derived from the same initial arbitrary polygon mesh by sequentially combining different subdivision rules. This added freedom allows for the more efficiently modeling of objects in computer graphics including objects and characters with semi-sharp features.

CROSS-REFERENCE TO RELATED APPLICATIONS

The following co-pending U.S. patent applications relate to the presentapplication. Each of the listed co-pending applications is assigned tothe same assignee as the present application.

1. U.S. patent application Ser. No. 08/905,436, filed Aug. 4, 1997 andtitled: “REALISTIC SURFACE SIMULATION IN COMPUTER ANIMATION”.

2. U.S. patent application Ser. No. 08/905,434, filed Aug. 4, 1997 andtitled: “TEXTURE MAPPING AND OTHER USES OF SCALAR FIELDS ON SUBDIVISONSURFACES IN COMPUTER GRAPHICS AND ANIMATION”.

FIELD OF THE INVENTION

The invention relates generally to the art of computer graphics and themodeling of objects. More particularly the invention relates to themodeling of objects using subdivision surfaces and to the modeling ofobjects with semi-sharp creases or edges.

BACKGROUND OF THE INVENTION

No real objects have perfectly sharp features. All edges, even those onseemingly sharp objects, have some finite radius of curvature. Thoughinevitable in real life, this unavoidable lack of precision presents adifficult problem for computer graphics and computer animation. Somemodeling methods, e.g., the use of piecewise linear surfaces (polygonmeshes) work well for objects with sharp boundaries. Other methods,e.g., NURBS (non-uniform rational B-splines) work well (i.e., are moreaccurate and compact) for modeling curved surfaces, but fair less welland are less efficient for modeling objects with sharp features.

In recent work, Hoppe, et al. have shown that piecewise smooth surfacesincorporating sharp features, including edges, creases, darts andcomers, can be efficiently modeled using subdivision surfaces byaltering the standard Loop subdivision rules in the region of such sharpfeatures. Hoppe, et al., Piecewise Smooth Surface Reconstruction,Computer Graphics (SIGGRAPH ‘94 Proceedings), pgs. 295-302. The modifiedsubdivision surface technique developed by Hoppe, et al, provides for anefficient method for modeling objects containing both curved surfacesand sharp features. The resulting sharp features are, however,infinitely sharp, i.e., the tangent plane is discontinuous across thesharp feature.

Even the method of Hoppe, et al. does not, therefore, solve the problemof modeling real objects with finite radius edges, creases and corners.To model such objects with existing techniques, either subdivisionsurfaces, NURBS, or polygons, requires vastly complicating the model byincluding many closely spaced control points or polygons in the regionof the finite radius contour. As soon as one moves away from infinitesharpness, which can be modeled easily and efficiently with subdivisionsurfaces following Hoppe, et al. most, if not all of the advantages ofthe method are lost, and one must create a vastly more complicated modelto enjoy the incremental enhancement in realism. Accordingly, there is aneed for a way to efficiently model objects with semi-sharp featuresand, more generally, there is a need for a way to sculpt the limitsurface defined by subdivision without complicating the initial mesh.

SUMMARY OF THE INVENTION

The present invention involves a method for modeling objects withsurfaces created by sequentially combining different subdivision rules.By subdividing a mesh a finite number of times with one or more sets of“special rules” before taking the infinite subdivision limit with“standard rules” one can produce different limit surfaces from the sameinitial mesh.

One important application of the invention is to the modeling of objectswith smooth and semi-sharp features. The present invention allows one tomodel objects with edges and creases of continuously variable sharpnesswithout adding vertices or otherwise changing the underlying controlpoint mesh. The result is achieved in the exemplary embodiment byexplicitly subdividing the initial mesh using the Hoppe, et al. rules ora variation which like the Hoppe, et al. rules do not require tangentplain continuity in the region of a sharp feature. After a finite numberof iterations, one switches to the traditional continuous tangent planerules for successive iterations. On may then push the final mesh pointsto their smooth surface infinite subdivision limits. The number ofiterations performed with the “sharp” (i.e., discontinuous tangentplane) rules determines the sharpness of the feature on the limitsurface. Though the number of explicit subdivisions must be an integer,“fractional smoothness” can be achieved by interpolating the position ofpoints between the locations determined using N and N+1 iterations withthe sharp rules.

In another exemplary embodiment, the invention is used to improve theshape of Catmull-Clark limit surfaces derived from initial meshes thatinclude triangular faces.

BRIEF DESCRIPTION OF THE DRAWINGS

The file of this patent contains at least one drawing executed in color.Copies of this patent with color drawings will be provided by the Patentand Trademark Office upon request and payment of the necessary fee.

FIG. 1 shows generally the elements of a computer system suitable forcarrying out the present invention.

FIG. 2 shows the main steps in generating a computer animation of acharacter using the techniques of the present invention.

FIG. 3 shows the control point mesh of Geri's head.

FIG. 4 shows some points on the skin mesh which contribute to the energyfunction.

FIG. 5 shows Geri's head with a scalar field for k_(h) in red.

FIG. 6 shows a portion of Geri's hand with articulation weights set tothe default value before smoothing.

FIG. 7 shows Geri's hand with articulation weights as scalar fieldvalues derived by smoothing.

FIG. 8 shows the result of texture mapping on a subdivision surfaceusing the method of the present invention.

FIG. 9 shows a neighborhood of a point in the control mesh prior tosubdivision.

FIG. 10 shows a region of the control mesh with a sharp crease and itssubdivision.

FIG. 11 shows the control mesh of Geri's hand with sharpness 1 featureshighlighted.

FIG. 12 shows the resulting semi-sharp features in a rendered image.

FIG. 13 shows a triangular face in the initial mesh and its firstsubdivision.

DETAILED DESCRIPTION OF AN EXEMPLARY EMBODIMENT

FIG. 1 shows a computer system suitable for carrying out the invention.A main bus 1 is connected to one or more CPU's 2 and a main memory 3.Also connected to the bus are a keyboard 4 and large disk memory 5. Theframe buffer 6 receives output information from the main bus and sendsit through another bus 7 to either a CRT or another peripheral whichwrites the image directly onto film. To illustrate the present inventionwe will describe its use in the animation of a character, Geri. The mainsteps in that process are summarized in FIG. 2.

The first step 21 involves the production of a two dimensional controlmesh corresponding to a character to be modeled and animated. In thepresent case, the mesh, shown in FIG. 3, was derived from data scannedin from a physical model of Geri's head. Alternatively, the mesh couldbe created internally through a CAD program without a physical model.Depending on the type of geometrical primitives to be used, e.g.polygons, NURBS or sub-division surfaces, the set of the control pointsshould be selected so that the chosen primitive representation providesan accurate fit to the original model or design and captures all posesof the character during the animation.

In an exemplary embodiment, Geri's head, depicted in FIG. 3, the controlmesh contains approximately 4100 control points grouped to formapproximately 4300 polygon faces. The points were entered manually usinga touch probe on a physical model. Much care was taken in selecting thecontrol points. Regions with fine scale details require more controlpoints, i.e., smaller polygons. In addition, regions with largerpolygons—greater spaced control points—should not directly abut regionsof small polygons, instead, to prevent artifacts in the rendered imagethe number and position of control points should be chosen so thattransitions between regions of high and low detail (i.e., little and bigpolygons) are smooth. In addition, one should select the control pointsso that creases, or other semi-sharp features lie along polygon edges(the treatment of these features will be discussed in detail below).Control points are stored as a list which includes an identifier and thethree dimensional coordinates for each point. Points are grouped aspolygons and stored in another list with a polygon identifier and theidentifiers of the included points. Sharp features are also stored asordered lists of vertices and sharpness values.

Once the set of control points, polygons and creases defining thekinematic head are entered and stored in the computer, the articulationengineer must determine how each point is to move for each gesture orcharacter movement. This step 22 is done by coding animation controls,which effectuate transformations of the model corresponding to differentmovements, e.g., jaw down, or left eyebrow up. The transformations maybe either linear displacements or, as in the “jaw down” control,rotation about a specified axis. The magnitude of the control, i.e., theamount of displacement or angle of rotation is specified by a scalarparameter, s, supplied by the animator. The corresponding transformationof the model can be written as a function T(s). The possible movementsof Geri's face are specified using approximately 252 such controls. Thearticulation engineer can specify the relative motion of differentcontrol points under a given animation control by assigning weights tothe control points. In the case of Geri's head, each of the 252animation controls affects the motion of approximately 150 controlpoints to which non-zero weights are assigned. The effect of the weightis to specify the “strength” of the transformation at the given point,i.e., the transformation T with strength s at a point with weight w isgiven by T(w*s). The weights for a given control can either be assignedindividually, control point by control point, or as a smooth scalarfield covering the portion of the mesh whose location is altered by thetransformation. The scalar field approach, described in detail in below,offers a great improvement in efficiency over the point-by-pointapproach which, for 252 controls times an average of 150 weights percontrol, would require individually specifying almost 40,000 weights tospecify the motion of Geri's head alone.

The next step 23 in FIG. 2 is the creation of a second mesh of points,the skin mesh, which will typically be in one-to-one correspondence withthe control points of the kinematic head. This need not be the case,however. In some embodiments, a skin mesh is used with a greater numberof control points than are contained in the kinematic head and aprojection of the skin points to the kinematic head is defined so thateach skin point is “tied” to a unique point on the kinematic head (whichneed not be a control point). Other embodiments may have a skin meshthat is sparser than the underlying kinematic head. In that case, thesame projection strategy can be used to tie skin control points topoints on the kinematic head.

In the exemplary embodiment of Geri's head, the positions of the skinpoints were determined by copying the kinematic head control pointswhile Geri's face was in the “neutral pose”, which was taken to be theexpression on the model head when it was scanned in with the touchprobe. The “neutral pose” defines the equilibrium positions of the skinpoints, i.e., the position of Geri's features for which his skin meshcontrol points will exactly match the positions of the control points ofthe kinematic head.

In step 24, the articulation engineer specifies the properties of Geri'sskin and the degree to which it is constrained to follow the underlyingkinematic head. This is done by specifying the local parameters of anenergy function, the extrema of which determine the position of thequasi-static surface, which in the present example is the skin on Geri'shead. Though the details of the energy function used to simulate theskin on Geri's head is illustrative, variations are possible and will bereadily apparent to those of ordinary skill in the art. The presentinvention is thus in no way limited to the details of the describedimplementation. That said, the energy function used to model Geri's headtakes the form:$E = {{\sum\limits_{\underset{pairs}{edge}}{E_{s1}\left( {P_{i},P_{j}} \right)}} + {\sum\limits_{\underset{pairs}{{non}\text{-}{edge}}}{E_{s2}\left( {P_{i},P_{j}} \right)}} + {\sum\limits_{triplets}{E_{p}\left( {P_{i},P_{j},P_{k}} \right)}} + {\sum\limits_{faces}{{E_{d}\left( {P_{i},P_{j},P_{k},P_{l}} \right)}{\sum\limits_{points}{E_{h}\left( P_{i} \right)}}}}}$

The first term accounts for stretching of the skin surface and is givenby:

E _(si)(P ₁ , P ₂)=k _(s1)(|P ₁ −P ₂ |−R)²

where P₁ and P_(2,) as shown in FIG. 4, are the locations (in R³) of twonearest neighbor points connected by an edge of the mesh and R is thedistance between the points in the neutral pose. K_(si) is a springconstant which can either be specified globally or locally for each edgeas a combination of scalar field values for the two points, e.g.,

K _(s1)(P ₁ , P ₂)=S ₁(P ₁)+S ₁(P ₂)

where S₁(P₁) is a scalar field defined on the mesh by one of the Imethods described below. The second term includes contributions fromnearest neighbor points which are not connected by an edge, e.g., pointsacross a diagonal in a quadrilateral. E_(s2)(P₁, P₃) has the same formas the first term but with a potentially different spring constantK_(s2) which again may be constant through out the mesh or definedlocally by a second scalar field,

K _(s2)(P ₁ , P ₃)=S ₂(P ₁)+S ₂(P ₃).

The three point function

E _(p)(P ₁ , P ₃ , P ₃)=K _(p)(|(P ₃ −P ₂)/D ₁−(P ₂ −P ₁)/D ₂ |−R _(p))²

is included to penalized bending and includes a contribution from eachconnected triplet of points. D₁ is the distance in R³ between P₃ and P₂in the neutral pose similarly D₂ is the distance between P₂ and P₁ inthe neutral pose. R_(p) is the value of

|(P ₃ −P ₂)/D ₁−(P ₂ −P ₁)/D ₂|

when all three points are in their neutral pose positions (note thatthis is not generally zero because (P₃−P₂) and (P₂−P₁) are vectors inR³). The four point function, E_(d)(P₁, P₂, P₃, P₄)=K_(d)E_(s2)(P₁,P₃)E_(s2)(P₂, P₄) includes a contribution from each quadrilateral faceand is included to penalize skewing. The coefficient K_(d) may either bea constant throughout the mesh or a combination of scalar field valuesat the four vertices.

The last term in the energy function penalizes the skin points forstraying from their corresponding kinematic mesh points:

E _(h)(P _(s))=K _(h) |P _(s) −P _(k)|²

where P_(s) is a skin mesh point and P_(k) is its corresponding point inthe kinematic mesh. The spring constant K_(h) is generally given by ascalar field associated with either the dynamic skin mesh or underlyingkinematic mesh.

Defining Scalar Fields

Scalar fields were used at several points in the above described processin order to define smoothly varying parameters on either the kinematicmesh or skin mesh. These include the articulation weights which specifythe relative movement of kinematic mesh control points under thedifferent animation control transformations, as well as the variousspring constants and other locally variable parameters of the energyfunction. When modeling with subdivision surfaces, the control pointmesh is a polygonal mesh of arbitrary topology, on which one cannotdefine global surface coordinates and on which there is no “natural” or“intrinsic” smooth local parameterization. The absence of a localparameterization has slowed the adoption of subdivision surfaces as ameans for modeling objects in computer animation, in part, because itwas believed that a surface parametrization was necessary to definescalar fields and perform parametric shading, e.g., texture or surfacemapping.

One aspect of the current invention is a solution to this problem and amethod for consistently defining scalar fields on subdivision surfaces,including scalar fields which can be used as “pseudo-coordinates” toperform parametric shading. Three exemplary methods for definingsmoothly varying scalar fields on arbitrary polygonal meshes which canbe consistently mapped through the recursive subdivision process aredescribed below.

Painting

The first method for defining scalar fields is by painting them directlyonto the surface mesh. In this technique, an image of a portion of thesurface on which the field is to be defined is painted using a standardtwo-dimensional computer painting program, e.g., Amazon, the intensityof the color applied to the image is chosen to represent the desiredmagnitude of the scalar field on the corresponding portion of thepainted surface, e.g. if one wants the skin of Geri's forehead to moreclosely follow the movement of his kinematic head than the flesh in hisjowls, one would paint his forehead correspondingly darker when“applying” the field giving rise to the parameter k_(h) in the abovedescribed energy function.

The first step in the painting method is to perform a single subdivisionof the control point mesh in the region on which one intends to definethe scalar field in order to obtain a mesh with only quadrilateralfaces, as will always be the case after the first subdivision withCatmull-Clark rules (discussed in detail in the Subdivision sectionbelow). The faces in the once subdivided mesh are then numbered andseparately coordinatized with two dimensional coordinates u and vassigned to each vertex (thus the need for quadrilateral faces). Thesurface is then further subdivided one or more additional times so thatthe resulting mesh sufficiently approximates the smooth surface limit.The u, v, coordinates for each patch (face in the once subdivided mesh)are carried through these additional subdivisions and coordinatize newvertices within the faces of the once subdivided mesh. The image is thenrendered and painted with a two dimensional painting program such thatthe distribution and intensity of the applied color represents thedesired local magnitude of the particular scalar field being defined.The painted image is then scanned and patch numbers, u, v, coordinates,and color intensity are stored for each pixel. Each patch is thenfurther subdivided until a dense set of points is obtained within eachpatch including points with u, v coordinates close to those stored foreach pixel in the painted image. The scanned pixel color intensities arethen assigned as target scalar field values to the corresponding pointsin this further subdivided mesh.

Going back to the initial undivided mesh, one then assigns to itsvertices initial guesses for the scalar field values which one hopesafter several subdivisions will reproduce the target values. The mesh isthen subdivided to the level of refinement at which the target valueswere assigned with scalar field values calculated from the initialguesses using the same subdivision rules used to calculate vertexlocations in the refined mesh. The values of the scalar field at thetarget points resulting from subdividing the mesh with the initialguesses is then compared with the target values at the correspondingpoints. Differences from the target values are computed and averagedback, reversing the subdivision calculations, to find corrections to theinitial guesses for scalar field values at the vertices in theunsubdivided mesh. This comparison and correction process is iterateduntil convergence. The result is a smooth field that closelyapproximates the painted intensities defined for any level ofsubdivision including the smooth surface limit.

Smoothing

The second more direct, if less picturesque, method for defining scalarfields is through smoothing. One begins by specifying values of thedesired field explicitly at some boundary or known points in a region ofthe mesh and solving for the rest through an iterative relaxationapproach constraining the scalar field value of the vertices to be equalto the average of their nearest neighbors. FIG. 5 shows Geri's face withthe k_(h) (“glue” field) shown in red. Smoothing was used in applyingthe k_(h) field to his chin region. The use of this method to assignsmoothly varying articulation weights is illustrated in FIGS. 6 and 7.To assign articulation weights to control points in the transitionregion between Geri's thumb which will move with full strength, w=1under the control, and his palm which will not move at all, i.e., haveweight O under the control, one begins by enforcing these conditionsand, as shown in FIG. 4, by giving the control points in theintermediate region default values of 1. One then performs the iterativerelaxation smoothing calculation until one obtains the result shown inFIG. 5., a smooth interpolation of intermediate scalar field values overthe control points in the transition region.

Energy Method

Scalar fields can also be used as pseudo coordinates to enableparameteric shading, e.g., texture and surface mapping. One reason thatsubdivision surfaces have not be more widely adopted for modeling incomputer graphics is that the absence of a surface parameterization wasthought to prevent the use of texture maps. One can, however, utilizingone aspect of the present invention, define s and t scalar fields(“pseudo-coordinates”) on regions of the subdivision surface which canserve as local coordinates for texture mapping. One cannot, of course,trick topology, so these pseudo coordinate fields can only beconsistently defined in topologically flat regions of the surface. Onecan then patch these regions together either discontinuously (e.g., ifone wanted two different pieces of cloth to abut along a seem) orcontinuously (by requiring fields in overlapping regions to have commonvalues). Pseudo-coordinates need only be defined, and consistencyrequirements need only be enforced after the model has been animated andonly in regions in which the texture mapping is to be applied. Thesestatic pseudo-coordinate patching constraints are thus far easier todeal with and satisfy than the ubiquitous kinematic constraints requiredto model complex objects with NURB patches.

Though s and t fields can be defined using either the painting orsmoothing method described above, an elaboration of the smoothingmethod, the energy method, is useful for defining scalar fields to beused as pseudo-coordinates for texture mapping. To avoid unacceptabledistortions in the surface texture, the mapping between the surface inR³ and the s and t pseudo-coordinate parameterization of the texturesample in R² should be at least roughly isometric (i.e. preserve lengthsand angles). If one pictures the mapping of the two dimensional textureonto the surface as fitting a rubber sheet over the surface, one wantsto minimize the amount of stretching and puckering. If the mapping isonto a portion of the surface with curvature, a pucker free mapping isnot possible. A best fit, so to speak, can be achieved, however, byutilizing the first two terms in the energy function described aboverestricted to two dimensions. Though this technique for achieving a“best fit” approximate isometry is useful for texture mapping onto othercurved surfaces, including those defined by NURBS, as will be readilyapparent to one of ordinary skill in the art, its implementation will bedescribed in detail for the case of subdivision surfaces.

The aim is to insure that as closely as the geometry allows distancesbetween points measured along the surface embedded in R³ equal distancesmeasured between the same points in the flat two dimensional s and tpseudo-coordinates of the texture patch for all points on the surface towhich the texture mapping is to be applied. To achieve a goodapproximation to this best compromise solution, we begin by subdividingthe initial mesh several times in the region to be texture mapped. Thisprovides a fine enough mesh so that the distances between nearestneighbor vertices measured in R³ sufficiently approximate the distancebetween the same points measured along the surface. One then generallyorients the texture on the surface by assigning s and t pseudocoordinates to several control points. One then completes the mapping,finding pseudo-coordinate s and t values for the remaining meshvertices, by minimizing a two dimensional energy function over theregion of the mesh on which the texture is to be applied. An exemplaryenergy function contains the two point functions from the energyfunction defined above, i.e.,$E = {{\sum\limits_{\underset{pairs}{edge}}{E_{1}\left( {P_{i},P_{j}} \right)}} + {\sum\limits_{\underset{pairs}{{non}\text{-}{edge}}}{E_{2}\left( {P_{i},P_{j}} \right)}}}$

with as before

E _(i)(P ₁ , P ₂ )=k _(i)(|P ₁ −P ₂ |−R)²

but now R is the distance between the points as measured along thesurface (as approximated by adding distances in three space M along thesubdivided mesh) and |P₁−P₂| is the two dimensional Euclidean distancebetween the points' s and t pseudo-coordinates, i.e.,$\sqrt{\left( {s_{1} - s_{2}} \right)^{2} + \left( {t_{1} - t_{2}} \right)^{2}}.$

This energy function can be minimized using Newton's method or otherstandard numerical technique as described below for minimizing the fullenergy function to determine the position of skin points.

Off-Line Steps

At this point in the process all of the manual steps in the animationare finished. The remaining steps can all be performed “off-line” as aprecursor to the actual rendering of the scene. Because these steps donot require additional input by the animator and take a small fractionof the computation time required for rendering, they add little to thetime required for the overall animation project.

Minimizing the Energy Function to Determine the Position of Skin Points

The positions of the control points for the surface mesh are determinedby minimizing the energy function. This can be done iteratively usingNewton's method or other standard numerical technique. For each frame ofanimation, the location of the skin control points can be takeninitially to correspond with the positions of the control points of thekinematic mesh. A quadratic approximation to the energy function is thencomputed and minimized yielding approximate positions for the controlpoints for the surface mesh. These approximate positions are used in thenext quadratic approximation which is solved yielding more refinedapproximate positions. In practice, a few iterations generally suffice.

To drive the system to a definite solution in the situation in which theenergy function has a null space, an additional term is added to theenergy function for each point in the dynamic mesh which, for the nthiteration, takes the form αN|P^(n) _(i)−P_(i) ^(n-1)|². Where N is thenumber of successive quadratic approximation iterations performed perframe and P_(j) ^(n) is the position of the jth control point after niterations. Though the discussion of the exemplary ff embodiment to thispoint has dealt primarily with the quasi-static determination of thesurface control point locations, this degeneracy breaking term can beslightly modified to allow for the incorporation of dynamic effects aswell, e.g., momentum of the surface mesh. To do so one can replace theabove described term with one of the form

E _(m) =αN|P _(i) ^(n)−2P _(i) ^(n)2|².

One can then treat the iteration index as indicating time steps as welland update the underlying kinematic control point positions in stepsbetween frames. The incorporation of momentum in this manner may beuseful in some cases, e.g., to automatically incorporate jiggling offleshy characters.

Subdivision

Once the positions of the skin mesh vertices are determined byiteratively minimizing the energy function, the model can be refinedprior to rendering in order to produce a smoother image. One method forrefining the mesh is through recursive subdivision following the methodof Catmull and Clark. See E. Catmull and J. Clark. Recursively generatedB-Spline surfaces on arbitrary topological meshes. Computer AidedDesign, 10(6):350-355, 1978. One could apply other subdivisionalgorithms, e.g., Doo and Sabin, see D. Doo and M. Sabin. Behavior ofrecursive division surfaces near extraordinary points. Computer AidedDesign, 10(6):356-360, 1978, or Loop if one chose the initial mesh to betriangular rather than quadrilateral, see Charles T. Loop. Smoothsubdivision surfaces based on triangles. M. S. Thesis, Department ofMathematics, University of Utah, August 1987. In the current exampleCatmull-Clark subdivision was used in part because a quadrilateral meshbetter fits the symmetries of the objects being modeled. In the limit ofinfinite recursion, Catmull-Clark subdivision leads to smooth surfaceswhich retain the topology of the initial mesh and which are locallyequivalent to B-Splines except in the neighborhoods of a finite numberof exceptional points at which the initial mesh had a vertex or facewith other than 4 edges.

The details of constructing smooth surfaces (and approximations tosmooth surfaces) through subdivision is now well known in the art andwill only be discussed briefly for completeness and as background to thediscussion of hybrid subdivision below. FIG. 9 shows a region about avertex in a quadrilateral mesh. This may be either the initial skin meshor the mesh resulting from one or more subdivisions. The vertex has nedges. For a “normal point,” n=4. If n is not equal to 4, the point isone of a finite number of exceptional points which will exist in anymesh that is not topologically flat. The number of such points will notincrease with subdivision. FIG. 10 shows the same region overlaid withthe mesh resulting from a single subdivision. The positions of theresulting vertices which can be characterized as either vertex, edge, orface vertices, are linear combinations of the neighboring vertices inthe unsubdivided mesh: $\begin{matrix}{q_{i} = {\frac{1}{4}\quad \left( {Q_{i} + R_{i - 1} + R_{i} + S} \right)}} \\{r_{i} = {\frac{1}{4}\quad \left( {q_{i} + q_{i + 1} + R_{i} + S} \right)}} \\{s = {{\sum\limits_{i = 1}^{n}\quad {\left( {q_{i} + R_{i}} \right)/n^{2}}} + \frac{\left( {n - 2} \right)}{nS}}}\end{matrix}$

Where j runs from l to n, the number of edges leaving the vertex S.Capital letters indicate points in the original mesh and small lettersrepresent points in the next subdivision. The subdivision process can bewritten as a linear transformation taking the vector V=(S,R₁, . . . ,R_(n), Q₁, . . . , Q_(n))^(T) to the vector v=(s,r₁, . . . , r_(n), q₁,. . . q_(n))^(T) defined by the 2n+1 by 2n+1 square matrixM_(n):v=M_(n)V. For Catmull-Clark subdivision about an ordinary point,i.e., when n=4, using the subdivision formulas given above M_(n) takesthe form: $M_{4} = {{1/16}\quad \begin{pmatrix}9 & \frac{3}{2} & \frac{3}{2} & \frac{3}{2} & \frac{3}{2} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \\6 & 6 & 1 & 0 & 1 & 1 & 0 & 0 & 1 \\6 & 1 & 6 & 1 & 0 & 1 & 1 & 0 & 0 \\6 & 0 & 1 & 6 & 1 & 0 & 1 & 1 & 0 \\6 & 1 & 0 & 1 & 6 & 0 & 0 & 1 & 1 \\4 & 4 & 4 & 0 & 0 & 4 & 0 & 0 & 0 \\4 & 0 & 4 & 4 & 0 & 0 & 4 & 0 & 0 \\4 & 0 & 0 & 4 & 4 & 0 & 0 & 4 & 0 \\4 & 4 & 0 & 0 & 4 & 0 & 0 & 0 & 4\end{pmatrix}}$

In practice, the subdivision procedure is applied a finite number oftimes, depending on the level of detail required of the object which maybe specified explicitly or determined automatically as a function of thescreen area of the object in a given scene. Typically if one starts withan initial mesh with detail on the order of the control mesh for Geri'shead, 4 or 5 subdivisions will suffice to produce quadrilateral facescorresponding to a screen area in the rendered image of a small fractionof a pixel. Iterating the above formulas gives the positions and scalarfield values for the vertices in the finer subdivided mesh as a functionof the positions and scalar field values of the control mesh.

After a sufficient number of iterations to produce enough vertices andsmall enough faces for the desired resolution, the vertex points andtheir scalar field values can be pushed to the infinite iteration smoothsurface limit. In the case of Catmull-Clark surfaces, the smooth surfacelimit is a cubic spline everywhere but in the neighborhoods of thefinite number of extraordinary points in the first subdivision of theoriginal mesh. These limit values are easily determined by analyzing theeigenspectrum and eigenvectors of the subdivison matrix, M_(n). See M.Halstead, M. Kass, and T. DeRose. Efficient, fair interpolation usingCatmull-Clark surfaces, Computer Graphics (SIGGRAPH 1993 Proceedings),volume 27, pages 35-44, August 1993. Specifically, applying the matrixM_(n) to a vector an infinite number of times projects out the portionof the initial vector along the direction of the dominant righteigenvector of the matrix M_(n). If we take m=2n+1, the size of thesquare matrix M_(n), then we can write the eigenvalues of M_(n) ase₁≧e₂>. . . e_(m) with corresponding right eigenvectors E₁, . . . ,E_(m). We can then uniquely decompose V as

V=c ₁ E ₁ +c ₂ E ₂ +c _(m) E _(m)

Where the c_(j) are three dimensional coordinates and/or values ofvarious scalar fields on the surface but are scalars in the vector spacein which M_(n) acts. Applying M_(n) we get M _(n) V=e ₁ c ₁ E ₁ +e ₂ c ₂E ₂ +. . . e _(m) c _(m) E _(m). For (M_(n))^(∞)V to have a nontriviallimit the largest eigenvalue of M_(n) must equal 1. So that(M_(n))^(∞)V=c₁E₁. Finally, Affine invariance requires that the rows ofM_(n) sum to 1 which means that E₁=(1 , . . . , 1). Which gives s^(∞=c)₁.

If one chooses a basis for the left eigenvectors of M_(n), L₁, . . .L_(m) so that they form an orthonormal set with their rightcounterparts, i.e., L_(j)·E_(k)=δ_(jk), this projection is given by thedot product L₁·V where L₁ is the left eigenvector of M_(n) witheigenvalue 1 and V is the column vector defined above of points in theneighborhood of S. For Catmull-Clark subdivision, the value of the thisdot product and thus the position of the point s after infinitesubdivision is given by$s^{\infty} = {\frac{1}{n\left( {n + 5} \right)}{\left( {{n^{2}s} + {4\quad {\sum\limits_{i = 1}^{n}r_{i}}} + {\sum\limits_{i = 1}^{n}q_{i}}} \right).}}$

This formula gives not only the smooth surface limit value of theposition of the vertex points in R³ but can also be used to calculatethe smooth surface limit of any scalar fields at those points as afunction of their values at points in the neighborhood of S.

Similarly, with a bit more math, see Halstead, et al. cited above, itcan be shown that the eigenvectors for the second and third largesteigenvalues of the local subdivision matrix M_(n) span the tangent spaceof the limit surface at the point s^(∞). One can again project these outof the vector V using the orthonormality property of the left and righteigenvectors giving

c ₂ =L ₂ ·V and c ₃ =L ₃ ·V.

Because c₂ and c₃ span the tangent space at the point s^(∞), their crossproduct gives a vector normal to the limit surface at that point, i.e.,

N ^(∞) =c ₂ ×c ₃

at the point s^(∞). These tangent vectors have been calculated forCatmull-Clark subdivision and are given by:$c_{2} = {{\sum\limits_{i = 1}^{n}{A_{n}{\cos \left( {2\quad \pi \quad {i/n}} \right)}r_{i}}} + {\left( {{\cos \left( {2\quad \pi \quad {i/n}} \right)} + {\cos \quad \left( {2\quad \pi \quad {\left( {i + 1} \right)/n}} \right)}} \right)q_{i}}}$

Where$A_{n} = {1 + {\cos \left( {2\quad {\pi/n}} \right)} + {{\cos \left( {\pi/n} \right)}\sqrt{2\left( {9 + {\cos \left( {2\quad {\pi/n}} \right)}} \right.}}}$

and c₃ is obtained by replacing r_(i) with r_(i+1) and q_(i) with q₁₊₁.

After constructing N^(∞), the smooth surface limit normal, at each ofthe vertices of the refined mesh. One can then interpolate these normalsacross the subpixel faces of the refined mesh to render the characterusing Phong shading.

Hybrid Subdivision Schemes

One aspect of the present invention is a method for creating surfaces bysequentially combining different subdivision rules. By subdividing amesh a finite number of times with one or more sets of “special rules”before taking the infinite subdivision limit with the “standard rules”,one can produce different limit surfaces from the same initial mesh.Combining different subdivision rules thus provides an additional degreeof freedom which can be a more efficient means for obtaining the desiredlimit surface than adding points to the initial mesh. Below we describein detail two particular exemplary uses of this aspect of the inventionin modeling Geri. Many others will be readily apparent to those ofordinary skill in the art.

Semi-Sharp Edges and Creases

Human skin is not a completely smooth surface. Faces and hands, forexample, contain somewhat sharp features, both creases and edges. Tomodel these features using the general smooth-surface-limit subdivisiontechniques outlined above, or more conventional B-spline patch methods,would require a very complicated initial control mesh in which thevertex spacing in regions of sharp features would have to be very small,leading to lots of points and quadrilaterals. One method for modelingsharp features with subdivision surfaces without adding substantially tothe complexity of the initial control mesh is to alter the subdivisionrules for points lying on corners or sharp edges or creases. Asdescribed above, these features can be incorporated and tagged in theinitial control mesh.

The sharpness of these features can be maintained throughout theotherwise smoothing subdivision process by decoupling the subdivisionprocess along sharp edges so that points on either side of the sharpfeature do not contribute to the position of points on the edge insubsequent subdivisions. Locally modifying the subdivision algorithm inthis way results in a limit surface with sharp edges or creases acrosswhich the tangent plane is discontinuous. Details of this method formodeling various sharp features using Loop subdivision on a triangularmesh can be found in Hoppe, et al. Piecewise smooth surfacereconstruction. Computer Graphics (SIGGRAPH ‘94 Proceedings) (1994)295-302.

The problem with using the Hoppe, et al. approach for the realisticanimation of characters is that skin surfaces do not have infinitelysharp edges. All real objects, including humans, have some finite radiusof curvature along their otherwise sharp features. This fact may beignored when rendering machined metal objects but is very apparent whenmodeling and rendering human faces. The exemplary embodiment includes amodification of the above described subdivision algorithm to allow themodeling and rendering of edges and creases of arbitrary andcontinuously variable sharpness.

FIG. 1 shows a portion of the initial control mesh of Geri's hand withsharp features highlighted. As described above, care was taken in theinitial choice of mesh vertices so that these sharp features lay alongedges of the initial mesh. Their position in the mesh was stored alongwith a sharpness value between 0 and 5 (assuming 5 explicit subdivisionswas the maximum performed before pushing the vertices to their limits).Because the skin mesh has the same topology as the control mesh, thecorresponding edge in the skin mesh is uniquely identified. To model anedge or crease with sharpness N where N is an integer, one applies thesharp edge subdivision technique described in Hoppe et al. to determinethe locations of points along the designated edge or crease for thefirst N subdivisions. Thus to determine new edge and vertex locationsalong the crease or edge instead of applying the standard Catmull-Clarkformulas described above, one uses for example the following sharp edgeformulas:

r _(i)=½R _(i)+½S

s=⅛R _(j)+⅛R _(k)+¾S

where R_(j) and R_(k) are the points on either side of S along the sharpedge or crease, as shown in FIG. 10. If S is a vertex at which a creaseor edge ceases to be sharp its position after subdivision is calculatedusing the normal Catmull-Clark formulas.

After N subdivisions one applies the normal smooth algorithm forsubsequent explicit subdivisions and then pushes the points to theirsmooth surface limits. Thus for sharpness 2 features, one applies thesharp formulas for the first two subdivisions and then uses the smoothformulas for subsequent subdivision and for taking the smooth surfacelimit. One can get “fractional sharpness” e.g., 1.5, by linearlyinterpolating edge point locations between the sharpness 1 and 2results, i.e., calculate the position of control points after performingone sharp and one smooth subdivision and calculate the position ofcontrol points after subdividing twice with the sharp subdivision rulesand take the average of the two locations before subdividing fartherwith the smooth rules or pushing the points to their smooth surfacelimits.

One can also determine the positions and tangents on the limit surfaceof points in the initial mesh on a semisharp feature without explicitlyiterating the semisharp rules, by analyzing the eigenstructure of thecombined transformation matrix in a manner similar to that describedabove for the case of smooth subdivision. The limit surface position forthe neighborhood of a semisharp feature of sharpness k is given byv^(∞)=(M_(smooth))^(∞)(M_(sharp))^(k)V where we have suppressed the nindex indicating the number of edges and added a subscript designatingsmooth and sharp transformations. As above for the smooth case, we canreplace the application of M_(smooth) an infinite number or times withdotting by L₁ its dominant left eigenvector, giving

v ^(∞) =L ₁·(M _(sharp))^(k) V.

We can then replace L₁ by its decomposition in terms of lefteigenvectors of M_(sharp).

L ₁ =t ₁ l ₁ +t ₂ l ₂ +. . . t _(m) l _(m)

where l_(i) are left eigenvectors of M_(sharp) and t_(j) are expansioncoefficients, scalars under M_(sharp). Applying M_(sharp) times from theright to this expansion, we get:

v ^(∞)=(t _(1δ) ₁ ^(k) l ₁ +. . . t _(m)δ_(m) ^(k) l _(m))

where the δ_(j) are the eigenvalues of M_(sharp).

One may also create a feature with varying sharpness along its length.Assuming again that the initial mesh is created so that the sharpfeature lies along one or more connected edges, one can specifysharpness values at the vertices of the feature either by hand or by oneof the scalar field techniques described above, limited to onedimension. One can calculate the limit values of the sharpness fieldtreating the sharp feature as a subdivision curve and taking its limitvalue using the dominant left eigenvector as described above for twodimensional surface subdivision. For uniform cubic B-Spline subdivisionalong a curve, that vector is given by u₁=⅙[1,4,1]. See, e.g, E.Stollnitz, T. DeRose and D. Salesin. Wavelets for Computer Graphics,1996, 61-72. The limit value of the field on a point along thesemi-sharp feature is thus given by ⅔ of its value plus one ⅙ of thevalue of each of its nearest neighbors.

The location of each vertex point on the semisharp feature is thencalculated using the limit value of its sharpness field, i.e., oneapplies the sharp rules in the neighborhood of that point the number oftimes specified by the point's sharpness value (including interpolation,as described above for non-integer sharpness values).

This continuous smoothing of sharp edges can be used in conjunction withother subdivision schemes as well including those proposed by Loop andDoo and Sabin, cited above. In particular, all of the sharp edgetechniques and results of Hoppe, et al. can be extended to allow for theefficient modeling of semi-sharp features by combining smooth and sharpsubdivision rules in the manner described above for Catmull-Clarksubdivision.

Improving The Surface In the Neighborhood of Triangular Faces

A second application of hybrid subdivision used in the exemplaryembodiment involves a modification to the standard Catmull-Clark rulesused to determine the location of face points in the first subdivisionof a triangular face. As described above, if one begins with anarbitrary polygon mesh which includes non-quadrilateral faces, after onesubdivision with Catmull-Clark rules, all faces in the refined mesh willbe quadrilateral (though some vertices will have more than four edges).If one has an initial mesh which is convex with some triangular faces,e.g., the initial control point mesh of Geri's head, the smooth surfaceresulting from standard Catmull-Clark subdivision will be somewhat“lumpy” in regions in which there were triangular faces in the originalmesh. The shape of the limit surface can be improved if when calculatingthe location of the face point for a triangular face in the firstsubdivision (the only time there will be triangular faces), one takesits location to be the average of the locations of its surrounding edgepoints after subdivision rather than the average of the its surroundingvertices in the undivided mesh, the standard Catmull-Clark rule. This isillustrated in FIG. 13. The solid lines and points P_(i) are points inthe initial mesh. The dashed lines show the once subdivided mesh withpoints e_(i) and f. Under normal Catmull-Clark rules the location of fis given by:

f=⅓(P ₁ +P ₂ +P ₃).

Instead, we take

f=⅓(e ₁ +e ₂ +e ₃).

Because the e_(i) depend the locations of neighboring faces as well asthe P_(i), f will be drawn closer to the center of a convex mesh and thelimit surface will have a smoother contour.

The specific arrangements and methods described herein are merelyillustrative of the principles of the present invention. Numerousmodifications in form and detail may be made by those of ordinary skillin the art without departing from the scope of the present invention.Although this invention has been shown in relation to particularembodiments, it should not be considered so limited. Rather, the presentinvention is limited only by the scope of the appended claims.

What is claimed is:
 1. A computer-implemented method for creating one ormore semi-sharp features in a graphical object represented, at least inpart as a polygonal mesh, said computer-implemented method comprising:a) storing a model of the graphical object as an initial polygon meshwherein said one or more semi-sharp features corresponds to one or moreedges of sad initial polygon mesh; b) creating an intermediate mesh bysubdividing one or more times said initial mesh in the region of thedesired semi-sharp feature wherein the location of points on thesubdivided one or more edges corresponding to the desired semi-sharpfeature depend only on the locations of points on the one or more edgesof said initial polygon mesh corresponding to the desired semisharpfeature; c) subdividing said intermediate mesh one or more times in theregion of the desired semi-sharp feature so that the location of pointsalong the subdivided edges corresponding to the desired semi-sharpfeature depend on the location of some points in the intermediate meshnot located on the edges corresponding to the desired semi-sharp featurein the intermediate mesh; and d) outputting a model of said graphicalobject comprising said desired semi-sharp feature.
 2. Acomputer-implemented method of creating one or more semi-sharp featuresin a graphical object represented, at least in part, as a polygonalmesh: a) storing a model of the graphical object as an initial polygonmesh wherein said one or more semi-sharp features corresponds to one ormore edges of said initial mesh; b) creating an intermediate mesh bysubdividing said initial mesh one or more times using a subdivisionalgorithm in the region of the desired semi-sharp feature which ifapplied an infinite number of times would result in a surface having adiscontinuous tangent plane in the region of the desired semi-sharpfeature. c) creating a final mesh by subdividing said intermediate meshone or more times using a subdivision algorithm in the region of thedesired semi-sharp feature which if applied an infinite number of timeswould result in surface with no tangent plane discontinuities; and d)outputting said final mesh.
 3. The method of claim 1 or 2 comprising thefurther step of determining the smooth surface limit locations of thepoints in said final mesh.
 4. The method of claim 3 comprising thefurther step of determining the smooth surface limit normals at thelimit locations of the points in the final mesh in order to shade therendered image.